Consider the sound wave travelling in ideal gases of \(\mathrm{He}, \mathrm{CH}_4\), and \(\mathrm{CO}_2\). All the gases have the same ratio \(\frac{P}{\rho}\), where \(P\) is the pressure and \(\rho\) is the density. The ratio of the speed of sound through the gases \(\mathrm{v}_{\mathrm{He}}: \mathrm{v}_{\mathrm{CH}_4}: \mathrm{v}_{\mathrm{CO}_2}\) is given by

A positive ion \(A\) and a negative ion \(B\) has charges \(6.67 \times 10^{-19} \mathrm{C}\) and \(9.6 \times 10^{-10} \mathrm{C}\), and masses \(19.2 \times 10^{-27} \mathrm{~kg}\) and \(9 \times 10^{-27} \mathrm{~kg}\) respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is \(P \times 10^{45}\), where the value of 10\(P\) is _______.
(Take \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-1}\) and universal gravitational constant as \(6.67 \times 10^{-11} \mathrm{Nm}^2 \mathrm{~kg}^{-2}\) )
Assume that charge may not be an integral multiple of electrons.

A single slit of width \(a\) is illuminated by a monochromatic light of wavelength \(600\,nm\). The value of \(a\) for which first minimum appears at \(\theta=30^{\circ}\) on the screen will be ........... \(\mu m\)

The current flowing through \(R _2\) is:

When a car is approaching the observer, the frequency of horn is \(100 Hz\). After passing the observer, it is \(50\,Hz\). If the observer moves with the car, the frequency will be \(\frac{ x }{3} Hz\) where \(x =.....\)

The height of victoria falls is \(63\, {m}\). What is the difference in temperature of water at the top and at the bottom of fall ? (In \(^{\circ} {C}\)) [Given \(1\, {cal}=4.2\, {J}\) and specific heat of water \(\left.=1 \,{cal} \,{g}^{-1}\;{ }^{\circ}\, {C}^{-1}\right]\)

An object viewed from a near point distance of \(25\, {cm}\), using a microscopic lens with magnification \('6',\) gives an unresolved image. A resolved image is observed at infinite distance with a total magnification double the earlier using an eyepiece along with the given lens and a tube of length \(0.6\, {m}\), if the focal length of the eyepiece is euqal to \(.... \,{cm} .\)

If the capacitance of a nanocapacitor is measured in terms of a unit \(u\) made by combining the electric charge \(e,\) Bohr radius \(a_0,\) Planck's constant \(h\) and speed of light \(c\) then

An \(emf\) of \(0.08\,V\) is induced in a metal rod of length \(10\,cm\) held normal to a uniform magnetic field of \(0.4\,T\), when moves with a velocity of ...........\(ms^{-1}\)

Two billiard balls of equal mass \(30 \,{g}\) strike a rigid wall with same speed of \(108\, {kmph}\) (as shown) but at different angles. If the balls get reflected with the same speed then the ratio of the magnitude of impulses imparted to ball \('a'\) and ball \('b'\) by the wall along \('X'\) direction is :

Three rods of Copper, Brass and Steel are welded together to form a \(Y\) shaped structure. Area of cross - section of each rod \(= 4\ cm^2\) . End of copper rod is maintained at \(100^o C \) where as ends ofbrass and steel are kept at \(0^o C\). Lengths of the copper, brass and steel rods are \(46, 13\) and \(12\ cms\) respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are \(0.92, 0.26\) and \(0.12\ CGS\) units respectively. Rate ofheat flow through copper rod is ....... \(cal\, s^{-1}\)